8 research outputs found
Fourier expansion in variational quantum algorithms
The Fourier expansion of the loss function in variational quantum algorithms
(VQA) contains a wealth of information, yet is generally hard to access. We
focus on the class of variational circuits, where constant gates are Clifford
gates and parameterized gates are generated by Pauli operators, which covers
most practical cases while allowing much control thanks to the properties of
stabilizer circuits. We give a classical algorithm that, for an -qubit
circuit and a single Pauli observable, computes coefficients of all
trigonometric monomials up to a degree in time bounded by
. Using the general structure and implementation of the
algorithm we reveal several novel aspects of Fourier expansions in
Clifford+Pauli VQA such as (i) reformulating the problem of computing the
Fourier series as an instance of multivariate boolean quadratic system (ii)
showing that the approximation given by a truncated Fourier expansion can be
quantified by the norm and evaluated dynamically (iii) tendency of
Fourier series to be rather sparse and Fourier coefficients to cluster together
(iv) possibility to compute the full Fourier series for circuits of non-trivial
sizes, featuring tens to hundreds of qubits and parametric gates.Comment: 10+5 pages, code available at https://github.com/idnm/FourierVQA,
comments welcom
Conformal symmetry in quasi-free Markovian open quantum systems
Conformal symmetry governs the behavior of closed systems near second-order
phase transitions, and is expected to emerge in open systems going through
dissipative phase transitions. We propose a framework allowing for a manifest
description of conformal symmetry in open Markovian systems. The key difference
from the closed case is that both conformal algebra and the algebra of local
fields are realized on the space of superoperators. We illustrate the framework
by a series of examples featuring systems with quadratic Hamiltonians and
linear jump operators, where the Liouvillian dynamics can be efficiently
analyzed using the formalism of third quantization. We expect that our
framework can be extended to interacting systems using an appropriate
generalization of the conformal bootstrap.Comment: 15 pages, supplementary Wolfram Mathematica notebook available at
https://github.com/idnm/third_quantization v2: minor revision (references
added, typos corrected) v2: Minor revisions done and typos correcte
Demonstration of a parity-time symmetry breaking phase transition using superconducting and trapped-ion qutrits
Scalable quantum computers hold the promise to solve hard computational
problems, such as prime factorization, combinatorial optimization, simulation
of many-body physics, and quantum chemistry. While being key to understanding
many real-world phenomena, simulation of non-conservative quantum dynamics
presents a challenge for unitary quantum computation. In this work, we focus on
simulating non-unitary parity-time symmetric systems, which exhibit a
distinctive symmetry-breaking phase transition as well as other unique features
that have no counterpart in closed systems. We show that a qutrit, a
three-level quantum system, is capable of realizing this non-equilibrium phase
transition. By using two physical platforms - an array of trapped ions and a
superconducting transmon - and by controlling their three energy levels in a
digital manner, we experimentally simulate the parity-time symmetry-breaking
phase transition. Our results indicate the potential advantage of multi-level
(qudit) processors in simulating physical effects, where additional accessible
levels can play the role of a controlled environment.Comment: 14 pages, 9 figure
Electromagnetic sources beyond common multipoles
The complete dynamic multipole expansion of electromagnetic sources contains more types of multipole terms than is conventionally perceived. The toroidal multipoles are one of the examples of such contributions that have been widely studied in recent years. Here we inspect more closely the other type of commonly overlooked terms known as the mean-square radii. In particular, we discuss both quantitative and qualitative aspects of the mean-square radii and provide a general geometrical framework for their visualization. We also consider the role of the mean-square radii in expanding the family of nontrivial nonradiating electromagnetic sources
Efficient variational synthesis of quantum circuits with coherent multi-start optimization
We consider the problem of the variational quantum circuit synthesis into a
gate set consisting of the CNOT gate and arbitrary single-qubit (1q) gates with
the primary target being the minimization of the CNOT count. First we note that
along with the discrete architecture search suffering from the combinatorial
explosion of complexity, optimization over 1q gates can also be a crucial
roadblock due to the omnipresence of local minimums (well known in the context
of variational quantum algorithms but apparently underappreciated in the
context of the variational compiling). Taking the issue seriously, we make an
extensive search over the initial conditions an essential part of our approach.
Another key idea we propose is to use parametrized two-qubit (2q) controlled
phase gates, which can interpolate between the identity gate and the CNOT gate,
and allow a continuous relaxation of the discrete architecture search, which
can be executed jointly with the optimization over 1q gates. This coherent
optimization of the architecture together with 1q gates appears to work
surprisingly well in practice, sometimes even outperforming optimization over
1q gates alone (for fixed optimal architectures). As illustrative examples and
applications we derive 8 CNOT and T depth 3 decomposition of the 3q Toffoli
gate on the nearest-neighbor topology, rediscover known best decompositions of
the 4q Toffoli gate on all 4q topologies including a 1 CNOT gate improvement on
the star-shaped topology, and propose decomposition of the 5q Toffoli gate on
the nearest-neighbor topology with 48 CNOT gates. We also benchmark the
performance of our approach on a number of 5q quantum circuits from the
ibm_qx_mapping database showing that it is highly competitive with the existing
software. The algorithm developed in this work is available as a Python package
CPFlow.Comment: 12+4 pages, code available at https://github.com/idnm/cpflow,
comments are welcome
Nonradiating anapole condition derived from Devaney-Wolf theorem and excited in a broken-symmetry dielectric particle
In this work, we first derive the nonradiating anapole condition with a straightforward theoretical demonstration exploiting one of the Devaney-Wolf theorems for nonradiating currents. Based on the equivalent volumetric and surface electromagnetic sources, it is possible to establish a unique compact conditions directly from Maxwell's Equations in order to ensure nonradiating anapole state. In addition, we support our theoretical findings with a numerical investigation on a broken-symmetry dielectric particle, building block of a metamaterial structure, demonstrating through a detailed multiple expansion the nonradiating anapole condition behind these peculiar destructive interactions. (C) 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreemen